{"id":"hexagon/2611.00025","pdf":"https://hexagonmath.org/pdf/2611.00025v1","url":"https://hexagonmath.org/abs/2611.00025v1","tags":["msc:11T06","msc:11T55","Cyclotomic Polynomials","Mersenne prime polynomials","finite fields","perfect polynomials","sum of divisors"],"score":0,"title":"The invariant $c_M$ of Gallardo and Rahavandrainy at Mersenne prime exponents, and the case $p=7$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["David Niedbala Giraudin"],"reviews":0,"subject":"math.NT","summary":"A Mersenne prime polynomial is an irreducible polynomial over $\\mathbb{F}_2$ of the form $x^a(x+1)^b+1$ with $a,b\\ge1$. Gallardo and Rahavandrainy conjectured that, apart from two exceptions, $\\sigma(M^{2h})$ always has an irreducible factor which is not a Mersenne prime. For a prime $p\\ge5$ they proved it for every $M$ outside an exceptional set $\\Sigma_p=\\Sigma_p^1\\cup\\dots\\cup\\Sigma_p^4$, defined through an integer $c_M$ and an odd integer $m_M$ read off the leading coefficients of $\\sigma(\\sigma(M^{p-1}))$, and later they proved it for every Mersenne prime number $p\\ge31$; their method…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-03","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}